Learning the joint distribution via random-order autoregressive modeling—A unified probabilistic framework for inertial confinement fusion design

In the inertial confinement fusion (ICF) community, AI-assisted design is highly anticipated but currently relies on a fragmented toolkit: surrogate forward models, inverse Bayesian inference tools, automated design optimizations, etc., each trained for a single specific direction of information flow. Yet all these tasks share a common mathematical structure—they query conditional distributions of the same underlying joint distribution p(design, performance). We propose to learn this joint distribution directly. Design parameters and performance metrics are treated as tokens, and a transformer is trained with random-order masking to approximate their joint probability density. Once trained, the diverse tasks mentioned above all reduce to a single operation: evaluating a conditional probability under the learned distribution. Applied to a dataset of 81 920 one-dimensional implosion simulations, the model reproduces well-established ICF scaling laws—validating that it has internalized the correct physical constraints—and proposes new designs under up to eight stringent constraints with a high success rate, high diversity, and a ten-thousand-fold efficiency gain over brute-force screening. In high-dimensional sub-regions of the design space where only a handful of training samples survive, the model still returns the correct and well-normalized conditional distributions, demonstrating that it captures the underlying manifold structure of physical constraints rather than merely interpolating between data points. Our framework features a simple architecture and a firm probabilistic theoretic foundation. It provides a new approach for analyzing strongly coupled systems like the ICF implosion.